Kinetic effects on the ideal-wall limit and the resistive wall mode
Jon Menard (PPPL)Z. Wang, Y. Liu (CCFE), S. Kaye, J.-K. Park,
J. Berkery (CU), S. Sabbagh (CU)and the NSTX Research Team
18th Workshop on MHD Stability ControlNovember 18, 2013
Santa Fe, New Mexico
NSTX-U Supported by
Culham Sci CtrYork U
Chubu UFukui U
Hiroshima UHyogo UKyoto U
Kyushu UKyushu Tokai U
NIFSNiigata UU Tokyo
JAEAInst for Nucl Res, Kiev
Ioffe InstTRINITI
Chonbuk Natl UNFRI
KAISTPOSTECH
Seoul Natl UASIPP
CIEMATFOM Inst DIFFER
ENEA, FrascatiCEA, Cadarache
IPP, JülichIPP, Garching
ASCR, Czech Rep
Coll of Wm & MaryColumbia UCompXGeneral AtomicsFIUINLJohns Hopkins ULANLLLNLLodestarMITLehigh UNova PhotonicsOld DominionORNLPPPLPrinceton UPurdue USNLThink Tank, Inc.UC DavisUC IrvineUCLAUCSDU ColoradoU IllinoisU MarylandU RochesterU TennesseeU TulsaU WashingtonU WisconsinX Science LLC
*This work supported by US DoE contract DE-AC02-09CH11466
NSTX-U 2013 Mode Control Meeting - Menard
Pressure-driven kink limit is strong physics constraint on maximum fusion performance
• Modes grow rapidly above kink limit:– g ~ 1-10% of tA
-1 where tA ~ 1ms
• Superconducting “ideal wall” can increase stable bN up to factor of 2
• Real wall resistive slow-growing “resistive wall mode” (RWM)– g twall ~ 1 – ms instead of ms time-scales
• RWM can be stabilized with:– kinetic effects (rotation, dissipation)– active feedback control
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Pfusion n2v p2 bT2 BT
4 bN4 BT
4 (1+k2)2 / A fBS2
Talk focuses on ideal-wall mode (IWM), also treats RWM vs. Wf
M. Chu, et al., Plasma Phys. Control. Fusion 52 (2010) 123001
NSTX-U 2013 Mode Control Meeting - Menard
Background
• Characteristic growth rates and frequencies of RWM and IWM– RWM: gtwall ~ 1 and wtwall < 1– IWM: gtA ~ 1-10% (gtwall >> 1) and wtA ~ WftA (1-30%) (wtwall >> 1)
• Kinetic effects important for RWM (J. Berkery invited TI2.02, Thu AM)– Publications: Berkery, et al. PRL 104 (2010) 035003, Sabbagh, et al., NF 50 (2010) 025020
• Rotation and kinetic effects largely unexplored for IWM– Such effects generally higher-order than fluid terms (p, J||, |dB|2, wall)
• Calculations for NSTX indicate both rotation and kinetic effects can modify both IWM and RWM stability limits– High toroidal rotation generated by co-injected NBI in NSTX
• Fast core rotation: Wf / wsound up to ~1, Wf / wAlfven ~ up to 0.1-0.3– Fluid/kinetic pressure is dominant instability drive in high-b ST plasmas
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NSTX-U 2013 Mode Control Meeting - Menard
Study 3 classes of IWM-unstable plasmas spanning low to high bN
• Low bN limit ~3.5, often saturated/long-lived mode– qmin ~ 2-3 – Common in early phase of current flat-top– Higher fraction of beam pressure, momentum (lower ne)
• Intermediate bN limit ~ 5– qmin ~1.2-1.5– Typical good-performance H-mode, H98 ~ 0.8-1.2
• Highest bN limit ~ 6-6.5– qmin ~ 1– “Enhanced Pedestal” H-mode high H98 ~ 1.5-1.6– Broad pressure, rotation profiles, high edge rotation shear 4
NSTX-U 2013 Mode Control Meeting - Menard
MARS-K: self-consistent linear resistive MHD including toroidal rotation and drift-kinetic effects
• Perturbed single-fluid linear MHD:
• Mode-particle resonance operator:
• Drift-kinetic effects in perturbed anisotropic pressure p:
• Fast ions: analytic slowing-down f(v) model – isotropic or anisotropic
Y.Q. Liu, et al., Phys. Plasmas 15, 112503 2008
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Precession ExB Transit and bounce Collisions
Diamagnetic
• Include toroidal flow only: vf = RWf(y) and wE = wE(y)
• Rotation and rotation shear effects:
This talk
NSTX-U 2013 Mode Control Meeting - Menard
Study 3 classes of IWM-unstable plasmas spanning low to high bN
• Low bN limit ~ 3.5, often saturated/long-lived– qmin ~ 2-3 – Common in early phase of current flat-top– Higher fraction of beam pressure, momentum (lower ne)
• Intermediate bN limit ~ 5– qmin ~1.2-1.5– Typical good-performance H-mode, H98 ~ 0.8-1.2
• Highest bN limit ~ 6-6.5– qmin ~ 1– “Enhanced Pedestal” H-mode high H98 ~ 1.5-1.6– Broad pressure, rotation profiles, high edge rotation shear 6
NSTX-U 2013 Mode Control Meeting - Menard
Saturated f=15-30kHz n=1 mode common during early IP flat-top phase
Mode clamps bN to ~3.5, reduces neutron rate ~20%
sometimes slows locks disrupts
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geff tA ~ 1×10-3
NSTX shot 138065
Fixed PNBI = 3MW, IP = 800kA, bT=10-15%
NSTX-U 2013 Mode Control Meeting - Menard
Fluid (non-kinetic) MARS-K calculations find:Rotation reduces IWL bN = 6 3-3.5
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Fluid MARS marginal bN ~ 3 – 4 consistent with experiment
NSTX shot 138065, t=376ms MARS n=1 in fluid limit
experimental value
High rotation bN limit ~ 4.5 3.2for Wf(0)tA = 10 17% (experimental)
Low rotation bN limit ~ 6for Wf(0)tA = 1 7%
Experiment
NSTX-U 2013 Mode Control Meeting - Menard
Kinetic mode also destabilized by rotation
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Kinetic mode tracked numerically by starting from fluid root and increasing kinetic fraction aK = 0 1 as G = 5/3 0
Non-adiabatic dWkinetic fraction varied (aK=0 1) including:
◄ No thermal or fast◄ Thermal and fast◄ Fast only◄ Thermal only
Experimental rotationbN = 5.6 < low-rotation IWL
g variation from fluid to kinetic mode can be non-monotonic Kinetic mode g fluid g
implies destabilization results from rotation
NSTX-U 2013 Mode Control Meeting - Menard
Kinetic stability limit similar to fluid limit:Marginal bN < 3.5 far below low-rotation bN limit of ~6
Convergence issues:• Drift-kinetic MHD model
assumes fluid G = 0
• When G = 0, MARS can sometimes track wrong root or find spurious root
• Solution: take G 0 limit, monitor eigenfunctions for continuous trend vs. ak, b, G
NSTX shot 138065, t=376ms Fluid and kinetic n=1
Experimental bN for n=1 mode onset10
Solid: Kinetic g
Dashed: Fluid g
NSTX-U 2013 Mode Control Meeting - Menard
Real part of complex energy functional consistent with rotational destabilization (dWrot ≤ 0) across minor radius
Coriolis - W
Coriolis - dW/dr
Centrifugal
Differential kinetic (always destabilizing)
NSTX shot 138065, t=376ms
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Destabilization from: Coriolis (dW/dr), centrifugal, differential kinetic
< 0
–
NSTX-U 2013 Mode Control Meeting - Menard
Study 3 classes of IWM-unstable plasmas spanning low to high bN
• Low bN limit ~ 3.5, often saturated/long-lived– qmin ~ 2-3 – Common in early phase of current flat-top– Higher fraction of beam pressure, momentum (lower ne)
• Intermediate bN limit ~ 5– qmin ~1.2-1.5– Typical good-performance H-mode, H98 ~ 0.8-1.2
• Highest bN limit ~ 6-6.5– qmin ~ 1– “Enhanced Pedestal” H-mode high H98 ~ 1.5-1.6– Broad pressure, rotation profiles, high edge rotation shear 12
NSTX-U 2013 Mode Control Meeting - Menard
Small f=30kHz continuous n=1 mode precedes larger 20-25kHz n=1 bursts
geff tA ~ 4×10-3First large n=1 burst
20% drop in bN
50% neutron rate dropLater n=1 modes full disruption
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NSTX-U 2013 Mode Control Meeting - Menard
Kinetic IWM bN limit consistent with experiment, fluid calculation under-predicts experimental limit
Fluid bN limit for low Wf
Kinetic bN limit for experimental Wf (weakly dependent on fast-ion peaking factor)
Fluid bN limit for experimental Wf (DbN = -1.4 vs. low rotation)Experimental bN for n=1 mode onset
NSTX shot 119621, t=610ms
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Fast-ion p profiles used in kinetic calculations
NSTX-U 2013 Mode Control Meeting - Menard
Measured IWM real frequency more consistent with kinetic model than fluid model
Fast-ion p profiles used in kinetic calculations
Measured mode frequencyNSTX shot 119621, t=610ms
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NSTX-U 2013 Mode Control Meeting - Menard
IWM: Kinetic fast-ions destabilizing, thermals stabilizing
Fast ions only: kinetic g exceeds all other g values
Thermals only: stabilizing
Kinetic g with no damping(non-adiabatic dWk = 0 limit)
Kinetic g with thermal + fast: similar to fluid g
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Implication: thermal damping stabilizes rotation-driven mode
NSTX-U 2013 Mode Control Meeting - Menard
IWM: Precession resonance dominates damping,highest bN requires inclusion of passing resonance
Passing resonance alone provides little stabilization: g gno-
damp
Precession resonance provides most stabilizationResonance occurs near location of wr = wE in core (not shown)
Passing more important than bounce resonance after precession
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no damping
NSTX-U 2013 Mode Control Meeting - Menard
High rotation reduces bN limits for ideal wall mode (IWM), no-wall mode (NWM), and resistive wall mode (RWM)
• At high rotation, RWM marginal bN limit can extend below fluid NWM limit, above fluid IWM limit, near kinetic IWM limit
Fluid limit (no kinetic effects) Fluid limit (no kinetic effects)
NSTX-U 2013 Mode Control Meeting - Menard
RWM: Rotation can change fluid RWM eigenfunction and move regions of singular displacement away from rationals
Solid: Wf(0)tA = 0.005, G=5/3 Dashed: Wf(0)tA = 0.20, G=0.0001
Note: bN values are not identical
NSTX-U 2013 Mode Control Meeting - Menard
Precession resonance alone cannot provide passive RWM stabilization – next step: include bounce harmonics, passing
• Thermals are destabilizing
• Fast ion contribution to stability is small
• Fast + thermal similar to thermal
• Precession + bounce calculations underway less destabilization
NSTX-U 2013 Mode Control Meeting - Menard
Study 3 classes of IWM-unstable plasmas spanning low to high bN
• Low bN limit ~ 3.5, often saturated/long-lived– qmin ~ 2-3 – Common in early phase of current flat-top– Higher fraction of beam pressure, momentum (lower ne)
• Intermediate bN limit ~ 5– qmin ~1.2-1.5– Typical good-performance H-mode, H98 ~ 0.8-1.2
• Highest bN limit ~ 6-6.5– qmin ~ 1– “Enhanced Pedestal” H-mode high H98 ~ 1.5-1.6– Broad pressure, rotation profiles, high edge rotation shear 21
NSTX-U 2013 Mode Control Meeting - Menard
Experimental characteristics of highest-bN MHD
• bN = 6-6.5 sustained for 2-3tE
– Oscillations from ELMs and bottom/limiter interactions
– Possible small RWM activity– Only small core MHD (steady
neutron rate)
geff tA ~ 1×10-2
• f = 50kHz mode causes 35% bN drop ending high-b phase– Mode grows very fast (~100ms)– n-number difficult to determine– Possible that mode has n > 1
NSTX shot 134991, t=760ms
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NSTX-U 2013 Mode Control Meeting - Menard
Kinetic IWM stability consistent with access to bN > 6Fluid calculation under-predicts experimental bN
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Fluid bN limit ~6.55 for low Wf
Kinetic bN limit ~ 6.3 for experimental Wf
Fluid bN limit ~ 5.7 for experimental Wf
Experimental bN range
Rotational de-stabilization weaker (DbN = -0.8 vs. low rotation)
NSTX shot 134991, t=760ms
NSTX-U 2013 Mode Control Meeting - Menard
High again rotation reduces bN limits for IWM, NWM, & RWM
• Fluid RWM bN limit can extend above the fluid IWM limit• Note: Fluid RWM bN limit is lower than kinetic IWM limit
– Possible operating window at very high bN where fluid RWM stable
Fluid limit (no kinetic effects) Fluid limit (no kinetic effects)
NSTX shot 134991, t=760ms
RWM G=10-4
NSTX-U 2013 Mode Control Meeting - Menard
Precession resonance alone provides RWM passive stabilization (passive stability consistent with experiment)
• Thermals marginally stabilizing: ak-crit ~ 0.75
• Fast ion contribution also stabilizing relative to fluid g
• Fast + thermal contributions strongly stabilizing: ak-crit ~ 0.02
NSTX-U 2013 Mode Control Meeting - Menard
IWM energy analysis near marginal stability elucidates trends from growth-rate scans
-5-4-3-2-1012345
Magnetic + pressure Vacuum Parallel Current Rotation
Low beta (138065)Mid beta (119621)High beta (134991)
Rotation WfParallel current J||Vacuump + dB bending and compression
• All cases: field-line bending+compression balances primarily p
• Low b: J|| (low q shear) and high Wf strongly destabilizing• Mid b: Reduced destabilization from J|| & Wf increases b
limit• High b: Large Wfꞌ at edge minimizes Wf drive highest b 26
Terms inRe(dW)dWvac-b
Note: terms should sum to ≈ zero
NSTX-U 2013 Mode Control Meeting - Menard
Summary• Rotation, kinetic effects can modify IWM & RWM at high Wf, b
– (From APS): Rotation effects most pronounced for plasmas near rotation-shear enhanced interchange/Kelvin-Helmholtz (KH) threshold
– High rotation shear near edge is most stable in theory and experiment• Kinetic damping from thermal resonances can be sufficient to
suppress rotation-driven IWM access low-rotation IWL• Fluid RWM b limits follow fluid NW and IW limits with rotation• Kinetic IWM b limits closer to experiment than fluid limits• Future work:– Understand kinetic damping of rotation-driven modes in more detail– Test more realistic fast-ion distribution functions – anisotropic / TRANSP– Assess finite orbit width effects (see next talk) – for fast, edge thermal ions– Assess modifications to RWM stability from rotation/rotation shear– Utilize off-axis NBI, NTV in NSTX-U to explore IWM, RWM limit vs. rotation
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NSTX-U 2013 Mode Control Meeting - Menard
Backup
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NSTX-U 2013 Mode Control Meeting - Menard
Analytic model of rotational-shear destabilization is being compared to MARS results and experiment
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Model: low rotation, high rotation shear (Ming Chu, Phys. Plasmas, Vol. 5, No. 1, (1998) 183)
> 0
2. Kelvin-Helmholtz criterion:bG = compressional Alfvén wave b
Ma = (shear) Alfvén wave excitation Mach number
Ms = sound Alfvén wave excitation Mach number
1. Ideal interchange criterion including rotation shear:
Ideal interchange index w/o rotationGlasser, Greene, Johnson – Phys. Fluids (1975) 875
>
Rotation shear destabilizes Alfvén wave directly or through sound wave coupling
NSTX-U 2013 Mode Control Meeting - Menard
Experiment marginally stable to rotation-driven interchange/Kelvin-Helmholtz near half-radius
Chu model marginal Wꞌ profiles:DI,W = 0 Ms
2 = bG DI,W (G=0) = 0p dominant KH dominant
Experimental Wꞌ profile
MARS full kinetic eigenfunction displacement largest near r/a~0.5
m=2 component dominant (qmin~2)
NSTX shot 138065, t=376ms
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NSTX-U 2013 Mode Control Meeting - Menard
Comparison of cases for KH marginal rotation shear
• Low b case– Rotation shear mode not
stabilized kinetically
• Medium b case– Nearly achieve no-rotation
IWL via kinetic stabilization
• High b case– High edge rotation shear
stabilizing – minimizes needed kinetic stabilization
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NSTX-U 2013 Mode Control Meeting - Menard
Inclusion of thermal and fast-ions (TRANSP) in total pressure can significantly modify pressure profile shape
rpol
Total pressure (thermal + fast)Reconstruction
(without kinetic pressure constraint)Thermal
Fast
Thermal + NBI (includes fast-ion angular momentum)
Thermal (C6+)
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NSTX shot 138065, t=376ms
NSTX-U 2013 Mode Control Meeting - Menard
Inclusion of fast-ion pressure and angular momentum (computed from TRANSP) significantly lowers marginal bN
• Increased pressure profile peaking from fast-ions lowers bN limit from 7.7 to 6.1 at low rotation
• Effective bN limit at experimental rotation reduced from 6.3 to ~3.4
NSTX shot 138065, t=376ms
Wf(0)tA = 0.13Wf(0)tA = 0.04
Wf(0)tA = 0.17Wf(0)tA = 0.02
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FLUID CALCULATIONS
NSTX-U 2013 Mode Control Meeting - Menard
Initial studies find anisotropic fast-ion distribution has damping vs. ak, G trends similar to isotropic
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AnisotropicIsotropic
Anisotropic fast-ion passing resonance can cause dWK singularities – investigating…
NSTX-U 2013 Mode Control Meeting - Menard
Real part of complex energy functional provides equation for growth-rate useful for understanding instability sources
Dispersion relation Kinetic energy Potential energy
Coriolis - W
Coriolis - dW/dr
Centrifugal
Differential kinetic
Growth rate equation: mode growth for dWre < 0
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Note: n = -1 in MARS